Senate is fine. There is enough support in Senate. The real hurdle is House of Representatives.
Senate and House are very different kind of animals and designed that way. Senators seek re-election once every six years as opposed to two years for reps. Also there are only 100 senators compared to 450 Reps. Senators can afford to take intellectual point of view. Reps. have keep be in tune with what common people are thinking and champion their causes as they have to seek re-election every two years. Also, Reps. represent much fewer number of constituents and are therefore relatively more accessible to 'm and therefore closer.
So effectively the Senate tends to think long-term and intellectually. And house makes sure that common people wishes are taken into account.
This effect is SO visible in the IRB. Like it or not, unless there is a change of heart in House Representatives, abandon all hopes. Senators and President can do all the song and dance they want, it'll come to nothing unless House concedes.